长篇文章：Why Do Americans Stink at Math?# NextGeneration - 我爱宝宝
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长篇文章：Why Do Americans Stink at Math?
When Akihiko Takahashi was a junior in college in 1978, he was like most of
the other students at his university in suburban Tokyo. He had a vague sense
of wanting to accomplish something but no clue what that something should b
e. But that spring he met a man who would become his mentor, and this relati
onship set the course of his entire career.
Takeshi Matsuyama was an elementary-school teacher, but like a small number
of instructors in Japan, he taught not just young children but also college
students who wanted to become teachers. At the university-affiliated element
ary school where Matsuyama taught, he turned his classroom into a kind of la
boratory, concocting and trying out new teaching ideas. When Takahashi met h
im, Matsuyama was in the middle of his boldest experiment yet — revolutioni
zing the way students learned math by radically changing the way teachers ta
ught it.
Continue reading the main story
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Instead of having students memorize and then practice endless lists of equat
ions — which Takahashi remembered from his own days in school — Matsuyama
taught his college students to encourage passionate discussions among childr
en so they would come to uncover math’s procedures, properties and proofs f
or themselves. One day, for example, the young students would derive the for
mula for finding the area of a rectangle; the next, they would use what they
learned to do the same for parallelograms. Taught this new way, math itself
seemed transformed. It was not dull misery but challenging, stimulating and
even fun.
Photo
Credit Photo illustration by Andrew B. Myers. Prop stylist: Randi Brookman H
arris.
Takahashi quickly became a convert. He discovered that these ideas came from
reformers in the United States, and he dedicated himself to learning to tea
ch like an American. Over the next 12 years, as the Japanese educational sys
tem embraced this more vibrant approach to math, Takahashi taught first thro
ugh sixth grade. Teaching, and thinking about teaching, was practically all
he did. A quiet man with calm, smiling eyes, his passion for a new kind of m
ath instruction could take his colleagues by surprise. “He looks very gentl
e and kind,” Kazuyuki Shirai, a fellow math teacher, told me through a tran
slator. “But when he starts talking about math, everything changes.”
Takahashi was especially enthralled with an American group called the Nation
al Council of Teachers of Mathematics, or N.C.T.M., which published manifest
oes throughout the 1980s, prescribing radical changes in the teaching of mat
h. Spending late nights at school, Takahashi read every one. Like many profe
ssionals in Japan, teachers often said they did their work in the name of th
eir mentor. It was as if Takahashi bore two influences: Matsuyama and the Am
erican reformers.
Takahashi, who is 58, became one of his country’s leading math teachers, on
ce attracting 1,000 observers to a public lesson. He participated in a class
room equivalent of “Iron Chef,” the popular Japanese television show. But
in 1991, when he got the opportunity to take a new job in America, teaching
at a school run by the Japanese Education Ministry for expats in Chicago, he
did not hesitate. With his wife, a graphic designer, he left his friends, f
amily, colleagues — everything he knew — and moved to the United States, e
ager to be at the center of the new math.
As soon as he arrived, he started spending his days off visiting American sc
hools. One of the first math classes he observed gave him such a jolt that h
e assumed there must have been some kind of mistake. The class looked exactl
y like his own memories of school. “I thought, Well, that’s only this clas
s,” Takahashi said. But the next class looked like the first, and so did th
e next and the one after that. The Americans might have invented the world’
s best methods for teaching math to children, but it was difficult to find a
nyone actually using them.
It wasn’t the first time that Americans had dreamed up a better way to teac
h math and then failed to implement it. The same pattern played out in the 1
960s, when schools gripped by a post-Sputnik inferiority complex unveiled an
ambitious “new math,” only to find, a few years later, that nothing actua
lly changed. In fact, efforts to introduce a better way of teaching math str
etch back to the 1800s. The story is the same every time: a big, excited pus
h, followed by mass confusion and then a return to conventional practices.
The trouble always starts when teachers are told to put innovative ideas int
o practice without much guidance on how to do it. In the hands of unprepared
teachers, the reforms turn to nonsense, perplexing students more than helpi
ng them. One 1965 Peanuts cartoon depicts the young blond-haired Sally strug
gling to understand her new-math assignment: “Sets . . . one to one matchin
g . . . equivalent sets . . . sets of one . . . sets of two . . . renaming t
wo. . . .” After persisting for three valiant frames, she throws back her h
ead and bursts into tears: “All I want to know is, how much is two and two?
”
Today the frustrating descent from good intentions to tears is playing out o
nce again, as states across the country carry out the latest wave of math re
forms: the Common Core. A new set of academic standards developed to replace
states’ individually designed learning goals, the Common Core math standar
ds are like earlier math reforms, only further refined and more ambitious. W
hereas previous movements found teachers haphazardly, through organizations
like Takahashi’s beloved N.C.T.M. math-teacher group, the Common Core has a
broader reach. A group of governors and education chiefs from 48 states ini
tiated the writing of the standards, for both math and language arts, in 200
9. The same year, the Obama administration encouraged the idea, making the a
doption of rigorous “common standards” a criterion for receiving a portion
of the more than $4 billion in Race to the Top grants. Forty-three states h
ave adopted the standards.
The opportunity to change the way math is taught, as N.C.T.M. declared in it
s endorsement of the Common Core standards, is “unprecedented.” And yet, o
nce again, the reforms have arrived without any good system for helping teac
hers learn to teach them. Responding to a recent survey by Education Week, t
eachers said they had typically spent fewer than four days in Common Core tr
aining, and that included training for the language-arts standards as well a
s the math.
Carefully taught, the assignments can help make math more concrete. Students
don’t just memorize their times tables and addition facts but also underst
and how arithmetic works and how to apply it to real-life situations. But in
practice, most teachers are unprepared and children are baffled, leaving pa
rents furious. The comedian Louis C.K. parodied his daughters’ homework in
an appearance on “The Late Show With David Letterman”: “It’s like, Bill
has three goldfish. He buys two more. How many dogs live in London?”
The inadequate implementation can make math reforms seem like the most absur
d form of policy change — one that creates a whole new problem to solve. Wh
y try something we’ve failed at a half-dozen times before, only to watch it
backfire? Just four years after the standards were first released, this arg
ument has gained traction on both sides of the aisle. Since March, four Repu
blican governors have opposed the standards. In New York, a Republican candi
date is trying to establish another ballot line, called Stop Common Core, fo
r the November gubernatorial election. On the left, meanwhile, teachers’ un
ions in Chicago and New York have opposed the reforms.
The fact that countries like Japan have implemented a similar approach with
great success offers little consolation when the results here seem so dreadf
ul. Americans might have written the new math, but maybe we simply aren’t s
uited to it. “By God,” wrote Erick Erickson, editor of the website RedStat
e, in an anti-Common Core attack, is it such “a horrific idea that we might
teach math the way math has always been taught.”
The new math of the ‘60s, the new new math of the ‘80s and today’s Common
Core math all stem from the idea that the traditional way of teaching math
simply does not work. As a nation, we suffer from an ailment that John Allen
Paulos, a Temple University math professor and an author, calls innumeracy
— the mathematical equivalent of not being able to read. On national tests,
nearly two-thirds of fourth graders and eighth graders are not proficient i
n math. More than half of fourth graders taking the 2013 National Assessment
of Educational Progress could not accurately read the temperature on a neat
ly drawn thermometer. (They did not understand that each hash mark represent
ed two degrees rather than one, leading many students to mistake 46 degrees
for 43 degrees.) On the same multiple-choice test, three-quarters of fourth
graders could not translate a simple word problem about a girl who sold 15 c
ups of lemonade on Saturday and twice as many on Sunday into the expression
“15 + (2×15).” Even in Massachusetts, one of the country’s highest-perfo
rming states, math students are more than two years behind their counterpart
s in Shanghai.
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The new math of the ’60s, the new, new math of the ’80s and today’s Commo
n Core math all stem from the idea that the traditional way of teaching math
simply does not work.
Adulthood does not alleviate our quantitative deficiency. A 2012 study compa
ring 16-to-65-year-olds in 20 countries found that Americans rank in the bot
tom five in numeracy. On a scale of 1 to 5, 29 percent of them scored at Lev
el 1 or below, meaning they could do basic arithmetic but not computations r
equiring two or more steps. One study that examined medical prescriptions go
ne awry found that 17 percent of errors were caused by math mistakes on the
part of doctors or pharmacists. A survey found that three-quarters of doctor
s inaccurately estimated the rates of death and major complications associat
ed with common medical procedures, even in their own specialty areas.
One of the most vivid arithmetic failings displayed by Americans occurred in
the early 1980s, when the A&W restaurant chain released a new hamburger to
rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W b
urger had more meat than the Quarter Pounder; in taste tests, customers pref
erred A&W’s burger. And it was less expensive. A lavish A&W television and
radio marketing campaign cited these benefits. Yet instead of leaping at the
great value, customers snubbed it.
Only when the company held customer focus groups did it become clear why. Th
e Third Pounder presented the American public with a test in fractions. And
we failed. Misunderstanding the value of one-third, customers believed they
were being overcharged. Why, they asked the researchers, should they pay the
same amount for a third of a pound of meat as they did for a quarter-pound
of meat at McDonald’s. The “4” in “?,” larger than the “3” in “?,”
led them astray.
But our innumeracy isn’t inevitable. In the 1970s and the 1980s, cognitive
scientists studied a population known as the unschooled, people with little
or no formal education. Observing workers at a Baltimore dairy factory in th
e ‘80s, the psychologist Sylvia Scribner noted that even basic tasks requir
ed an extensive amount of math. For instance, many of the workers charged wi
th loading quarts and gallons of milk into crates had no more than a sixth-g
rade education. But they were able to do math, in order to assemble their lo
ads efficiently, that was “equivalent to shifting between different base sy
stems of numbers.” Throughout these mental calculations, errors were “virt
ually nonexistent.” And yet when these workers were out sick and the dairy’
s better-educated office workers filled in for them, productivity declined.
The unschooled may have been more capable of complex math than people who we
re specifically taught it, but in the context of school, they were stymied b
y math they already knew. Studies of children in Brazil, who helped support
their families by roaming the streets selling roasted peanuts and coconuts,
showed that the children routinely solved complex problems in their heads to
calculate a bill or make change. When cognitive scientists presented the ch
ildren with the very same problem, however, this time with pen and paper, th
ey stumbled. A 12-year-old boy who accurately computed the price of four coc
onuts at 35 cruzeiros each was later given the problem on paper. Incorrectly
using the multiplication method he was taught in school, he came up with th
e wrong answer. Similarly, when Scribner gave her dairy workers tests using
the language of math class, their scores averaged around 64 percent. The cog
nitive-science research suggested a startling cause of Americans’ innumerac
y: school.
Most American math classes follow the same pattern, a ritualistic series of
steps so ingrained that one researcher termed it a cultural script. Some tea
chers call the pattern “I, We, You.” After checking homework, teachers ann
ounce the day’s topic, demonstrating a new procedure: “Today, I’m going t
o show you how to divide a three-digit number by a two-digit number” (I). T
hen they lead the class in trying out a sample problem: “Let’s try out the
steps for 242 ÷ 16” (We). Finally they let students work through similar
problems on their own, usually by silently making their way through a work s
heet: “Keep your eyes on your own paper!” (You).
By focusing only on procedures — “Draw a division house, put ‘242’ on th
e inside and ‘16’ on the outside, etc.” — and not on what the procedures
mean, “I, We, You” turns school math into a sort of arbitrary process who
lly divorced from the real world of numbers. Students learn not math but, in
the words of one math educator, answer-getting. Instead of trying to convey
, say, the essence of what it means to subtract fractions, teachers tell stu
dents to draw butterflies and multiply along the diagonal wings, add the ant
ennas and finally reduce and simplify as needed. The answer-getting strategi
es may serve them well for a class period of practice problems, but after a
week, they forget. And students often can’t figure out how to apply the str
ategy for a particular problem to new problems.
How could you teach math in school that mirrors the way children learn it in
the world? That was the challenge Magdalene Lampert set for herself in the
1980s, when she began teaching elementary-school math in Cambridge, Mass. Sh
e grew up in Trenton, accompanying her father on his milk deliveries around
town, solving the milk-related math problems he encountered. “Like, you kno
w: If Mrs. Jones wants three quarts of this and Mrs. Smith, who lives next d
oor, wants eight quarts, how many cases do you have to put on the truck?” L
ampert, who is 67 years old, explained to me.
She knew there must be a way to tap into what students already understood an
d then build on it. In her classroom, she replaced “I, We, You” with a str
ucture you might call “You, Y’all, We.” Rather than starting each lesson
by introducing the main idea to be learned that day, she assigned a single “
problem of the day,” designed to let students struggle toward it — first o
n their own (You), then in peer groups (Y’all) and finally as a whole class
(We). The result was a process that replaced answer-getting with what Lampe
rt called sense-making. By pushing students to talk about math, she invited
them to share the misunderstandings most American students keep quiet until
the test. In the process, she gave them an opportunity to realize, on their
own, why their answers were wrong.
Lampert, who until recently was a professor of education at the University o
f Michigan in Ann Arbor, now works for the Boston Teacher Residency, a progr
am serving Boston public schools, and the New Visions for Public Schools net
work in New York City, instructing educators on how to train teachers. In he
r book, “Teaching Problems and the Problems of Teaching,” Lampert tells th
e story of how one of her fifth-grade classes learned fractions. One day, a
student made a “conjecture” that reflected a common misconception among ch
ildren. The fraction 5 / 6, the student argued, goes on the same place on th
e number line as 5 / 12. For the rest of the class period, the student liste
ned as a lineup of peers detailed all the reasons the two numbers couldn’t
possibly be equivalent, even though they had the same numerator. A few days
later, when Lampert gave a quiz on the topic (“Prove that 3 / 12 = 1 / 4 ,”
for example), the student could confidently declare why: “Three sections o
f the 12 go into each fourth.”
Over the years, observers who have studied Lampert’s classroom have found t
hat students learn an unusual amount of math. Rather than forgetting algorit
hms, they retain and even understand them. One boy who began fifth grade dec
laring math to be his worst subject ended it able to solve multiplication, l
ong division and fraction problems, not to mention simple multivariable equa
tions. It’s hard to look at Lampert’s results without concluding that with
the help of a great teacher, even Americans can become the so-called math p
eople we don’t think we are.
Among math reformers, Lampert’s work gained attention. Her research was cit
ed in the same N.C.T.M. standards documents that Takahashi later pored over.
She was featured in Time magazine in 1989 and was retained by the producers
of “Sesame Street” to help create the show “Square One Television,” aim
ed at making math accessible to children. Yet as her ideas took off, she beg
an to see a problem. In Japan, she was influencing teachers she had never me
t, by way of the N.C.T.M. standards. But where she lived, in America, teache
rs had few opportunities for learning the methods she developed.
Photo
Credit Photo illustration by Andrew B. Myers. Prop stylist: Randi Brookman H
arris. Butterfly icon by Tim Boelaars.
American institutions charged with training teachers in new approaches to ma
th have proved largely unable to do it. At most education schools, the profe
ssors with the research budgets and deanships have little interest in the sc
ience of teaching. Indeed, when Lampert attended Harvard’s Graduate School
of Education in the 1970s, she could find only one listing in the entire cou
rse catalog that used the word “teaching” in its title. (Today only 19 out
of 231 courses include it.) Methods courses, meanwhile, are usually taught
by the lowest ranks of professors — chronically underpaid, overworked and,
ultimately, ineffective.
Without the right training, most teachers do not understand math well enough
to teach it the way Lampert does. “Remember,” Lampert says, “American te
achers are only a subset of Americans.” As graduates of American schools, t
hey are no more likely to display numeracy than the rest of us. “I’m just
not a math person,” Lampert says her education students would say with an a
pologetic shrug.
Consequently, the most powerful influence on teachers is the one most beyond
our control. The sociologist Dan Lortie calls the phenomenon the apprentice
ship of observation. Teachers learn to teach primarily by recalling their me
mories of having been taught, an average of 13,000 hours of instruction over
a typical childhood. The apprenticeship of observation exacerbates what the
education scholar Suzanne Wilson calls education reform’s double bind. The
very people who embody the problem — teachers — are also the ones charged
with solving it.
Lampert witnessed the effects of the double bind in 1986, a year after Calif
ornia announced its intention to adopt “teaching for understanding,” a sty
le of math instruction similar to Lampert’s. A team of researchers that inc
luded Lampert’s husband, David Cohen, traveled to California to see how the
teachers were doing as they began to put the reforms into practice. But aft
er studying three dozen classrooms over four years, they found the new teach
ing simply wasn’t happening. Some of the failure could be explained by acti
ve resistance. One teacher deliberately replaced a new textbook’s problem-s
olving pages with the old worksheets he was accustomed to using.
Continue reading the main story
Teachers primarily learn to teach by recalling their memories of having been
taught, about 13,000 hours of instruction during a typical childhood — a p
roblem since their instruction wasn’t very good.
Much more common, though, were teachers who wanted to change, and were willi
ng to work hard to do it, but didn’t know how. Cohen observed one teacher,
for example, who claimed to have incited a “revolution” in her classroom.
But on closer inspection, her classroom had changed but not in the way Calif
ornia reformers intended it to. Instead of focusing on mathematical ideas, s
he inserted new activities into the traditional “I, We You” framework. The
supposedly cooperative learning groups she used to replace her rows of desk
s, for example, seemed in practice less a tool to encourage discussion than
a means to dismiss the class for lunch (this group can line up first, now th
at group, etc.).
And how could she have known to do anything different? Her principal praised
her efforts, holding them up as an example for others. Official math-reform
training did not help, either. Sometimes trainers offered patently bad info
rmation — failing to clarify, for example, that even though teachers were t
o elicit wrong answers from students, they still needed, eventually, to get
to correct ones. Textbooks, too, barely changed, despite publishers’ claims
to the contrary.
With the Common Core, teachers are once more being asked to unlearn an old a
pproach and learn an entirely new one, essentially on their own. Training is
still weak and infrequent, and principals — who are no more skilled at mat
h than their teachers — remain unprepared to offer support. Textbooks, once
again, have received only surface adjustments, despite the shiny Common Cor
e labels that decorate their covers. “To have a vendor say their product is
Common Core is close to meaningless,” says Phil Daro, an author of the mat
h standards.
Continue reading the main story
RECENT COMMENTS
PA 2 hours ago
Amount of resources, Financial or effort that USA can put towards keeping it
s competitive edge, developing nations like India and China...
Nora Madsen 2 hours ago
We needed all this research, international comparisons and Takahashi's life
story to come to the conclusion that American math teachers need...
David Bee 2 hours ago
After reading the article and scores of Comments it was found surprising tha
t no one mentioned (other than Ms. Green, in a way --- see the...
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Left to their own devices, teachers are once again trying to incorporate new
ideas into old scripts, often botching them in the process. One especially
nonsensical result stems from the Common Core’s suggestion that students no
t just find answers but also “illustrate and explain the calculation by usi
ng equations, rectangular arrays, and/or area models.” The idea of utilizin
g arrays of dots makes sense in the hands of a skilled teacher, who can use
them to help a student understand how multiplication actually works. For exa
mple, a teacher trying to explain multiplication might ask a student to firs
t draw three rows of dots with two dots in each row and then imagine what th
e picture would look like with three or four or five dots in each row. Guidi
ng the student through the exercise, the teacher could help her see that eac
h march up the times table (3x2, 3x3, 3x4) just means adding another dot per
row. But if a teacher doesn’t use the dots to illustrate bigger ideas, the
y become just another meaningless exercise. Instead of memorizing familiar s
teps, students now practice even stranger rituals, like drawing dots only to
count them or breaking simple addition problems into complicated forms (62+
26, for example, must become 60+2+20+6) without understanding why. This can
make for even poorer math students. “In the hands of unprepared teachers,”
Lampert says, “alternative algorithms are worse than just teaching them st
andard algorithms.”
No wonder parents and some mathematicians denigrate the reforms as “fuzzy m
ath.” In the warped way untrained teachers interpret them, they are fuzzy.
When Akihiko Takahashi arrived in America, he was surprised to find how rare
ly teachers discussed their teaching methods. A year after he got to Chicago
, he went to a one-day conference of teachers and mathematicians and was per
plexed by the fact that the gathering occurred only twice a year. In Japan,
meetings between math-education professors and teachers happened as a matter
of course, even before the new American ideas arrived. More distressing to
Takahashi was that American teachers had almost no opportunities to watch on
e another teach.
In Japan, teachers had always depended on jugyokenkyu, which translates lite
rally as “lesson study,” a set of practices that Japanese teachers use to
hone their craft. A teacher first plans lessons, then teaches in front of an
audience of students and other teachers along with at least one university
observer. Then the observers talk with the teacher about what has just taken
place. Each public lesson poses a hypothesis, a new idea about how to help
children learn. And each discussion offers a chance to determine whether it
worked. Without jugyokenkyu, it was no wonder the American teachers’ work f
ell short of the model set by their best thinkers. Without jugyokenyku, Taka
hashi never would have learned to teach at all. Neither, certainly, would th
e rest of Japan’s teachers.
The best discussions were the most microscopic, minute-by-minute recollectio
ns of what had occurred, with commentary. If the students were struggling to
represent their subtractions visually, why not help them by, say, arranging
tile blocks in groups of 10, a teacher would suggest. Or after a geometry l
esson, someone might note the inherent challenge for children in seeing angl
es as not just corners of a triangle but as quantities — a more difficult s
tretch than making the same mental step for area. By the end, the teachers h
ad learned not just how to teach the material from that day but also about m
ath and the shape of students’ thoughts and how to mold them.
If teachers weren’t able to observe the methods firsthand, they could find
textbooks, written by the leading instructors and focusing on the idea of al
lowing students to work on a single problem each day. Lesson study helped th
e textbook writers home in on the most productive problems. For example, if
you are trying to decide on the best problem to teach children to subtract a
one-digit number from a two-digit number using borrowing, or regrouping, yo
u have many choices: 11 minus 2, 18 minus 9, etc. Yet from all these options
, five of the six textbook companies in Japan converged on the same exact pr
oblem, Toshiakira Fujii, a professor of math education at Tokyo Gakugei Univ
ersity, told me. They determined that 13 minus 9 was the best. Other problem
s, it turned out, were likely to lead students to discover only one solution
method. With 12 minus 3, for instance, the natural approach for most studen
ts was to take away 2 and then 1 (the subtraction-subtraction method). Very
few would take 3 from 10 and then add back 2 (the subtraction-addition metho
d).
But Japanese teachers knew that students were best served by understanding b
oth methods. They used 13 minus 9 because, faced with that particular proble
m, students were equally likely to employ subtraction-subtraction (take away
3 to get 10, and then subtract the remaining 6 to get 4) as they were to us
e subtraction-addition (break 13 into 10 and 3, and then take 9 from 10 and
add the remaining 1 and 3 to get 4). A teacher leading the “We” part of th
e lesson, when students shared their strategies, could do so with full confi
dence that both methods would emerge.
By 1995, when American researchers videotaped eighth-grade classrooms in the
United States and Japan, Japanese schools had overwhelmingly traded the old
“I, We, You” script for “You, Y’all, We.” (American schools, meanwhile
didn’t look much different than they did before the reforms.) Japanese stu
dents had changed too. Participating in class, they spoke more often than Am
ericans and had more to say. In fact, when Takahashi came to Chicago initial
ly, the first thing he noticed was how uncomfortably silent all the classroo
ms were. One teacher must have said, “Shh!” a hundred times, he said. Late
r, when he took American visitors on tours of Japanese schools, he had to wa
rn them about the noise from children talking, arguing, shrieking about the
best way to solve problems. The research showed that Japanese students initi
ated the method for solving a problem in 40 percent of the lessons; American
s initiated 9 percent of the time. Similarly, 96 percent of American student
s’ work fell into the category of “practice,” while Japanese students spe
nt only 41 percent of their time practicing. Almost half of Japanese student
s’ time was spent doing work that the researchers termed “invent/think.”
(American students spent less than 1 percent of their time on it.) Even the
equipment in classrooms reflected the focus on getting students to think. Wh
ereas American teachers all used overhead projectors, allowing them to focus
students’ attention on the teacher’s rules and equations, rather than the
ir own, in Japan, the preferred device was a blackboard, allowing students t
o track the evolution of everyone’s ideas.
Japanese schools are far from perfect. Though lesson study is pervasive in e
lementary and middle school, it is less so in high school, where the emphasi
s is on cramming for college entrance exams. As is true in the United States
, lower-income students in Japan have recently been falling behind their pee
rs, and people there worry about staying competitive on international tests.
Yet while the United States regularly hovers in the middle of the pack or b
elow on these tests, Japan scores at the top. And other countries now inchin
g ahead of Japan imitate the jugyokenkyu approach. Some, like China, do this
by drawing on their own native jugyokenkyu-style traditions (zuanyan jiaoca
i, or “studying teaching materials intensively,” Chinese teachers call it)
. Others, including Singapore, adopt lesson study as a deliberate matter of
government policy. Finland, meanwhile, made the shift by carving out time fo
r teachers to spend learning. There, as in Japan, teachers teach for 600 or
fewer hours each school year, leaving them ample time to prepare, revise and
learn. By contrast, American teachers spend nearly 1,100 hours with little
feedback.
It could be tempting to dismiss Japan’s success as a cultural novelty, an u
nreproducible result of an affluent, homogeneous, and math-positive society.
Perhaps the Japanese are simply the “math people” Americans aren’t. Yet
when I visited Japan, every teacher I spoke to told me a story that sounded
distinctly American. “I used to hate math,” an elementary-school teacher n
amed Shinichiro Kurita said through a translator. “I couldn’t calculate. I
was slow. I was always at the bottom of the ladder, wondering why I had to
memorize these equations.” Like Takahashi, when he went to college and saw
his instructors teaching differently, “it was an enlightenment.”
Learning to teach the new way himself was not easy. “I had so much trouble,
” Kurita said. “I had absolutely no idea how to do it.” He listened caref
ully for what Japanese teachers call children’s twitters — mumbled nuggets
of inchoate thoughts that teachers can mold into the fully formed concept t
hey are trying to teach. And he worked hard on bansho, the term Japanese tea
chers use to describe the art of blackboard writing that helps students visu
alize the flow of ideas from problem to solution to broader mathematical pri
nciples. But for all his efforts, he said, “the children didn’t twitter, a
nd I couldn’t write on the blackboard.” Yet Kurita didn’t give up — and
he had resources to help him persevere. He went to study sessions with other
teachers, watched as many public lessons as he could and spent time with hi
s old professors. Eventually, as he learned more, his students started to do
the same. Today Kurita is the head of the math department at Setagaya Eleme
ntary School in Tokyo, the position once held by Takahashi’s mentor, Matsuy
ama.
Of all the lessons Japan has to offer the United States, the most important
might be the belief in patience and the possibility of change. Japan, after
all, was able to shift a country full of teachers to a new approach. Telling
me his story, Kurita quoted what he described as an old Japanese saying abo
ut perseverance: “Sit on a stone for three years to accomplish anything.”
Admittedly, a tenacious commitment to improvement seems to be part of the Ja
panese national heritage, showing up among teachers, autoworkers, sushi chef
s and tea-ceremony masters. Yet for his part, Akihiko Takahashi extends his
optimism even to a cause that can sometimes seem hopeless — the United Stat
es. After the great disappointment of moving here in 1991, he made a decisio
n his colleagues back in Japan thought was strange. He decided to stay and t
ry to help American teachers embrace the innovative ideas that reformers lik
e Magdalene Lampert pioneered.
Today Takahashi lives in Chicago and holds a full-time job in the education
department at DePaul University. (He also has a special appointment at his a
lma mater in Japan, where he and his wife frequently visit.) When it comes t
o transforming teaching in America, Takahashi sees promise in individual Ame
rican schools that have decided to embrace lesson study. Some do this delibe
rately, working with Takahashi to transform the way they teach math. Others
have built versions of lesson study without using that name. Sometimes these
efforts turn out to be duds. When carefully implemented, though, they show
promise. In one experiment in which more than 200 American teachers took par
t in lesson study, student achievement rose, as did teachers’ math knowledg
e — two rare accomplishments.
Training teachers in a new way of thinking will take time, and American pare
nts will need to be patient. In Japan, the transition did not happen overnig
ht. When Takahashi began teaching in the new style, parents initially compla
ined about the young instructor experimenting on their children. But his ear
ly explorations were confined to just a few lessons, giving him a chance to
learn what he was doing and to bring the parents along too. He began sending
home a monthly newsletter summarizing what the students had done in class a
nd why. By his third year, he was sending out the newsletter every day. If t
hey were going to support their children, and support Takahashi, the parents
needed to know the new math as well. And over time, they learned.
To cure our innumeracy, we will have to accept that the traditional approach
we take to teaching math — the one that can be mind-numbing, but also comf
ortingly familiar — does not work. We will have to come to see math not as
a list of rules to be memorized but as a way of looking at the world that re
ally makes sense.
CONTINUE READING THE MAIN STORY
598
COMMENTS
The other shift Americans will have to make extends beyond just math. Across
all school subjects, teachers receive a pale imitation of the preparation,
support and tools they need. And across all subjects, the neglect shows in s
tudents’ work. In addition to misunderstanding math, American students also
, on average, write weakly, read poorly, think unscientifically and grasp hi
story only superficially. Examining nearly 3,000 teachers in six school dist
ricts, the Bill & Melinda Gates Foundation recently found that nearly two-th
irds scored less than “proficient” in the areas of “intellectual challeng
e” and “classroom discourse.” Odds-defying individual teachers can be fou
nd in every state, but the overall picture is of a profession struggling to
make the best of an impossible hand.
Most policies aimed at improving teaching conceive of the job not as a craft
that needs to be taught but as a natural-born talent that teachers either d
ecide to muster or don’t possess. Instead of acknowledging that changes lik
e the new math are something teachers must learn over time, we mandate them
as “standards” that teachers are expected to simply “adopt.” We shouldn’
t be surprised, then, that their students don’t improve.
Here, too, the Japanese experience is telling. The teachers I met in Tokyo h
ad changed not just their ideas about math; they also changed their whole co
nception of what it means to be a teacher. “The term ‘teaching’ came to m
ean something totally different to me,” a teacher named Hideto Hirayama tol
d me through a translator. It was more sophisticated, more challenging — an
d more rewarding. “The moment that a child changes, the moment that he unde
rstands something, is amazing, and this transition happens right before your
eyes,” he said. “It seems like my heart stops every day.”
When Akihiko Takahashi was a junior in college in 1978, he was like most of
the other students at his university in suburban Tokyo. He had a vague sense
of wanting to accomplish something but no clue what that something should b
e. But that spring he met a man who would become his mentor, and this relati
onship set the course of his entire career.
Takeshi Matsuyama was an elementary-school teacher, but like a small number
of instructors in Japan, he taught not just young children but also college
students who wanted to become teachers. At the university-affiliated element
ary school where Matsuyama taught, he turned his classroom into a kind of la
boratory, concocting and trying out new teaching ideas. When Takahashi met h
im, Matsuyama was in the middle of his boldest experiment yet — revolutioni
zing the way students learned math by radically changing the way teachers ta
ught it.
Continue reading the main story
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Instead of having students memorize and then practice endless lists of equat
ions — which Takahashi remembered from his own days in school — Matsuyama
taught his college students to encourage passionate discussions among childr
en so they would come to uncover math’s procedures, properties and proofs f
or themselves. One day, for example, the young students would derive the for
mula for finding the area of a rectangle; the next, they would use what they
learned to do the same for parallelograms. Taught this new way, math itself
seemed transformed. It was not dull misery but challenging, stimulating and
even fun.
Photo
Credit Photo illustration by Andrew B. Myers. Prop stylist: Randi Brookman H
arris.
Takahashi quickly became a convert. He discovered that these ideas came from
reformers in the United States, and he dedicated himself to learning to tea
ch like an American. Over the next 12 years, as the Japanese educational sys
tem embraced this more vibrant approach to math, Takahashi taught first thro
ugh sixth grade. Teaching, and thinking about teaching, was practically all
he did. A quiet man with calm, smiling eyes, his passion for a new kind of m
ath instruction could take his colleagues by surprise. “He looks very gentl
e and kind,” Kazuyuki Shirai, a fellow math teacher, told me through a tran
slator. “But when he starts talking about math, everything changes.”
Takahashi was especially enthralled with an American group called the Nation
al Council of Teachers of Mathematics, or N.C.T.M., which published manifest
oes throughout the 1980s, prescribing radical changes in the teaching of mat
h. Spending late nights at school, Takahashi read every one. Like many profe
ssionals in Japan, teachers often said they did their work in the name of th
eir mentor. It was as if Takahashi bore two influences: Matsuyama and the Am
erican reformers.
Takahashi, who is 58, became one of his country’s leading math teachers, on
ce attracting 1,000 observers to a public lesson. He participated in a class
room equivalent of “Iron Chef,” the popular Japanese television show. But
in 1991, when he got the opportunity to take a new job in America, teaching
at a school run by the Japanese Education Ministry for expats in Chicago, he
did not hesitate. With his wife, a graphic designer, he left his friends, f
amily, colleagues — everything he knew — and moved to the United States, e
ager to be at the center of the new math.
As soon as he arrived, he started spending his days off visiting American sc
hools. One of the first math classes he observed gave him such a jolt that h
e assumed there must have been some kind of mistake. The class looked exactl
y like his own memories of school. “I thought, Well, that’s only this clas
s,” Takahashi said. But the next class looked like the first, and so did th
e next and the one after that. The Americans might have invented the world’
s best methods for teaching math to children, but it was difficult to find a
nyone actually using them.
It wasn’t the first time that Americans had dreamed up a better way to teac
h math and then failed to implement it. The same pattern played out in the 1
960s, when schools gripped by a post-Sputnik inferiority complex unveiled an
ambitious “new math,” only to find, a few years later, that nothing actua
lly changed. In fact, efforts to introduce a better way of teaching math str
etch back to the 1800s. The story is the same every time: a big, excited pus
h, followed by mass confusion and then a return to conventional practices.
The trouble always starts when teachers are told to put innovative ideas int
o practice without much guidance on how to do it. In the hands of unprepared
teachers, the reforms turn to nonsense, perplexing students more than helpi
ng them. One 1965 Peanuts cartoon depicts the young blond-haired Sally strug
gling to understand her new-math assignment: “Sets . . . one to one matchin
g . . . equivalent sets . . . sets of one . . . sets of two . . . renaming t
wo. . . .” After persisting for three valiant frames, she throws back her h
ead and bursts into tears: “All I want to know is, how much is two and two?
”
Today the frustrating descent from good intentions to tears is playing out o
nce again, as states across the country carry out the latest wave of math re
forms: the Common Core. A new set of academic standards developed to replace
states’ individually designed learning goals, the Common Core math standar
ds are like earlier math reforms, only further refined and more ambitious. W
hereas previous movements found teachers haphazardly, through organizations
like Takahashi’s beloved N.C.T.M. math-teacher group, the Common Core has a
broader reach. A group of governors and education chiefs from 48 states ini
tiated the writing of the standards, for both math and language arts, in 200
9. The same year, the Obama administration encouraged the idea, making the a
doption of rigorous “common standards” a criterion for receiving a portion
of the more than $4 billion in Race to the Top grants. Forty-three states h
ave adopted the standards.
The opportunity to change the way math is taught, as N.C.T.M. declared in it
s endorsement of the Common Core standards, is “unprecedented.” And yet, o
nce again, the reforms have arrived without any good system for helping teac
hers learn to teach them. Responding to a recent survey by Education Week, t
eachers said they had typically spent fewer than four days in Common Core tr
aining, and that included training for the language-arts standards as well a
s the math.
Carefully taught, the assignments can help make math more concrete. Students
don’t just memorize their times tables and addition facts but also underst
and how arithmetic works and how to apply it to real-life situations. But in
practice, most teachers are unprepared and children are baffled, leaving pa
rents furious. The comedian Louis C.K. parodied his daughters’ homework in
an appearance on “The Late Show With David Letterman”: “It’s like, Bill
has three goldfish. He buys two more. How many dogs live in London?”
The inadequate implementation can make math reforms seem like the most absur
d form of policy change — one that creates a whole new problem to solve. Wh
y try something we’ve failed at a half-dozen times before, only to watch it
backfire? Just four years after the standards were first released, this arg
ument has gained traction on both sides of the aisle. Since March, four Repu
blican governors have opposed the standards. In New York, a Republican candi
date is trying to establish another ballot line, called Stop Common Core, fo
r the November gubernatorial election. On the left, meanwhile, teachers’ un
ions in Chicago and New York have opposed the reforms.
The fact that countries like Japan have implemented a similar approach with
great success offers little consolation when the results here seem so dreadf
ul. Americans might have written the new math, but maybe we simply aren’t s
uited to it. “By God,” wrote Erick Erickson, editor of the website RedStat
e, in an anti-Common Core attack, is it such “a horrific idea that we might
teach math the way math has always been taught.”
The new math of the ‘60s, the new new math of the ‘80s and today’s Common
Core math all stem from the idea that the traditional way of teaching math
simply does not work. As a nation, we suffer from an ailment that John Allen
Paulos, a Temple University math professor and an author, calls innumeracy
— the mathematical equivalent of not being able to read. On national tests,
nearly two-thirds of fourth graders and eighth graders are not proficient i
n math. More than half of fourth graders taking the 2013 National Assessment
of Educational Progress could not accurately read the temperature on a neat
ly drawn thermometer. (They did not understand that each hash mark represent
ed two degrees rather than one, leading many students to mistake 46 degrees
for 43 degrees.) On the same multiple-choice test, three-quarters of fourth
graders could not translate a simple word problem about a girl who sold 15 c
ups of lemonade on Saturday and twice as many on Sunday into the expression
“15 + (2×15).” Even in Massachusetts, one of the country’s highest-perfo
rming states, math students are more than two years behind their counterpart
s in Shanghai.
Continue reading the main story
The new math of the ’60s, the new, new math of the ’80s and today’s Commo
n Core math all stem from the idea that the traditional way of teaching math
simply does not work.
Adulthood does not alleviate our quantitative deficiency. A 2012 study compa
ring 16-to-65-year-olds in 20 countries found that Americans rank in the bot
tom five in numeracy. On a scale of 1 to 5, 29 percent of them scored at Lev
el 1 or below, meaning they could do basic arithmetic but not computations r
equiring two or more steps. One study that examined medical prescriptions go
ne awry found that 17 percent of errors were caused by math mistakes on the
part of doctors or pharmacists. A survey found that three-quarters of doctor
s inaccurately estimated the rates of death and major complications associat
ed with common medical procedures, even in their own specialty areas.
One of the most vivid arithmetic failings displayed by Americans occurred in
the early 1980s, when the A&W restaurant chain released a new hamburger to
rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W b
urger had more meat than the Quarter Pounder; in taste tests, customers pref
erred A&W’s burger. And it was less expensive. A lavish A&W television and
radio marketing campaign cited these benefits. Yet instead of leaping at the
great value, customers snubbed it.
Only when the company held customer focus groups did it become clear why. Th
e Third Pounder presented the American public with a test in fractions. And
we failed. Misunderstanding the value of one-third, customers believed they
were being overcharged. Why, they asked the researchers, should they pay the
same amount for a third of a pound of meat as they did for a quarter-pound
of meat at McDonald’s. The “4” in “?,” larger than the “3” in “?,”
led them astray.
But our innumeracy isn’t inevitable. In the 1970s and the 1980s, cognitive
scientists studied a population known as the unschooled, people with little
or no formal education. Observing workers at a Baltimore dairy factory in th
e ‘80s, the psychologist Sylvia Scribner noted that even basic tasks requir
ed an extensive amount of math. For instance, many of the workers charged wi
th loading quarts and gallons of milk into crates had no more than a sixth-g
rade education. But they were able to do math, in order to assemble their lo
ads efficiently, that was “equivalent to shifting between different base sy
stems of numbers.” Throughout these mental calculations, errors were “virt
ually nonexistent.” And yet when these workers were out sick and the dairy’
s better-educated office workers filled in for them, productivity declined.
The unschooled may have been more capable of complex math than people who we
re specifically taught it, but in the context of school, they were stymied b
y math they already knew. Studies of children in Brazil, who helped support
their families by roaming the streets selling roasted peanuts and coconuts,
showed that the children routinely solved complex problems in their heads to
calculate a bill or make change. When cognitive scientists presented the ch
ildren with the very same problem, however, this time with pen and paper, th
ey stumbled. A 12-year-old boy who accurately computed the price of four coc
onuts at 35 cruzeiros each was later given the problem on paper. Incorrectly
using the multiplication method he was taught in school, he came up with th
e wrong answer. Similarly, when Scribner gave her dairy workers tests using
the language of math class, their scores averaged around 64 percent. The cog
nitive-science research suggested a startling cause of Americans’ innumerac
y: school.
Most American math classes follow the same pattern, a ritualistic series of
steps so ingrained that one researcher termed it a cultural script. Some tea
chers call the pattern “I, We, You.” After checking homework, teachers ann
ounce the day’s topic, demonstrating a new procedure: “Today, I’m going t
o show you how to divide a three-digit number by a two-digit number” (I). T
hen they lead the class in trying out a sample problem: “Let’s try out the
steps for 242 ÷ 16” (We). Finally they let students work through similar
problems on their own, usually by silently making their way through a work s
heet: “Keep your eyes on your own paper!” (You).
By focusing only on procedures — “Draw a division house, put ‘242’ on th
e inside and ‘16’ on the outside, etc.” — and not on what the procedures
mean, “I, We, You” turns school math into a sort of arbitrary process who
lly divorced from the real world of numbers. Students learn not math but, in
the words of one math educator, answer-getting. Instead of trying to convey
, say, the essence of what it means to subtract fractions, teachers tell stu
dents to draw butterflies and multiply along the diagonal wings, add the ant
ennas and finally reduce and simplify as needed. The answer-getting strategi
es may serve them well for a class period of practice problems, but after a
week, they forget. And students often can’t figure out how to apply the str
ategy for a particular problem to new problems.
How could you teach math in school that mirrors the way children learn it in
the world? That was the challenge Magdalene Lampert set for herself in the
1980s, when she began teaching elementary-school math in Cambridge, Mass. Sh
e grew up in Trenton, accompanying her father on his milk deliveries around
town, solving the milk-related math problems he encountered. “Like, you kno
w: If Mrs. Jones wants three quarts of this and Mrs. Smith, who lives next d
oor, wants eight quarts, how many cases do you have to put on the truck?” L
ampert, who is 67 years old, explained to me.
She knew there must be a way to tap into what students already understood an
d then build on it. In her classroom, she replaced “I, We, You” with a str
ucture you might call “You, Y’all, We.” Rather than starting each lesson
by introducing the main idea to be learned that day, she assigned a single “
problem of the day,” designed to let students struggle toward it — first o
n their own (You), then in peer groups (Y’all) and finally as a whole class
(We). The result was a process that replaced answer-getting with what Lampe
rt called sense-making. By pushing students to talk about math, she invited
them to share the misunderstandings most American students keep quiet until
the test. In the process, she gave them an opportunity to realize, on their
own, why their answers were wrong.
Lampert, who until recently was a professor of education at the University o
f Michigan in Ann Arbor, now works for the Boston Teacher Residency, a progr
am serving Boston public schools, and the New Visions for Public Schools net
work in New York City, instructing educators on how to train teachers. In he
r book, “Teaching Problems and the Problems of Teaching,” Lampert tells th
e story of how one of her fifth-grade classes learned fractions. One day, a
student made a “conjecture” that reflected a common misconception among ch
ildren. The fraction 5 / 6, the student argued, goes on the same place on th
e number line as 5 / 12. For the rest of the class period, the student liste
ned as a lineup of peers detailed all the reasons the two numbers couldn’t
possibly be equivalent, even though they had the same numerator. A few days
later, when Lampert gave a quiz on the topic (“Prove that 3 / 12 = 1 / 4 ,”
for example), the student could confidently declare why: “Three sections o
f the 12 go into each fourth.”
Over the years, observers who have studied Lampert’s classroom have found t
hat students learn an unusual amount of math. Rather than forgetting algorit
hms, they retain and even understand them. One boy who began fifth grade dec
laring math to be his worst subject ended it able to solve multiplication, l
ong division and fraction problems, not to mention simple multivariable equa
tions. It’s hard to look at Lampert’s results without concluding that with
the help of a great teacher, even Americans can become the so-called math p
eople we don’t think we are.
Among math reformers, Lampert’s work gained attention. Her research was cit
ed in the same N.C.T.M. standards documents that Takahashi later pored over.
She was featured in Time magazine in 1989 and was retained by the producers
of “Sesame Street” to help create the show “Square One Television,” aim
ed at making math accessible to children. Yet as her ideas took off, she beg
an to see a problem. In Japan, she was influencing teachers she had never me
t, by way of the N.C.T.M. standards. But where she lived, in America, teache
rs had few opportunities for learning the methods she developed.
Photo
Credit Photo illustration by Andrew B. Myers. Prop stylist: Randi Brookman H
arris. Butterfly icon by Tim Boelaars.
American institutions charged with training teachers in new approaches to ma
th have proved largely unable to do it. At most education schools, the profe
ssors with the research budgets and deanships have little interest in the sc
ience of teaching. Indeed, when Lampert attended Harvard’s Graduate School
of Education in the 1970s, she could find only one listing in the entire cou
rse catalog that used the word “teaching” in its title. (Today only 19 out
of 231 courses include it.) Methods courses, meanwhile, are usually taught
by the lowest ranks of professors — chronically underpaid, overworked and,
ultimately, ineffective.
Without the right training, most teachers do not understand math well enough
to teach it the way Lampert does. “Remember,” Lampert says, “American te
achers are only a subset of Americans.” As graduates of American schools, t
hey are no more likely to display numeracy than the rest of us. “I’m just
not a math person,” Lampert says her education students would say with an a
pologetic shrug.
Consequently, the most powerful influence on teachers is the one most beyond
our control. The sociologist Dan Lortie calls the phenomenon the apprentice
ship of observation. Teachers learn to teach primarily by recalling their me
mories of having been taught, an average of 13,000 hours of instruction over
a typical childhood. The apprenticeship of observation exacerbates what the
education scholar Suzanne Wilson calls education reform’s double bind. The
very people who embody the problem — teachers — are also the ones charged
with solving it.
Lampert witnessed the effects of the double bind in 1986, a year after Calif
ornia announced its intention to adopt “teaching for understanding,” a sty
le of math instruction similar to Lampert’s. A team of researchers that inc
luded Lampert’s husband, David Cohen, traveled to California to see how the
teachers were doing as they began to put the reforms into practice. But aft
er studying three dozen classrooms over four years, they found the new teach
ing simply wasn’t happening. Some of the failure could be explained by acti
ve resistance. One teacher deliberately replaced a new textbook’s problem-s
olving pages with the old worksheets he was accustomed to using.
Continue reading the main story
Teachers primarily learn to teach by recalling their memories of having been
taught, about 13,000 hours of instruction during a typical childhood — a p
roblem since their instruction wasn’t very good.
Much more common, though, were teachers who wanted to change, and were willi
ng to work hard to do it, but didn’t know how. Cohen observed one teacher,
for example, who claimed to have incited a “revolution” in her classroom.
But on closer inspection, her classroom had changed but not in the way Calif
ornia reformers intended it to. Instead of focusing on mathematical ideas, s
he inserted new activities into the traditional “I, We You” framework. The
supposedly cooperative learning groups she used to replace her rows of desk
s, for example, seemed in practice less a tool to encourage discussion than
a means to dismiss the class for lunch (this group can line up first, now th
at group, etc.).
And how could she have known to do anything different? Her principal praised
her efforts, holding them up as an example for others. Official math-reform
training did not help, either. Sometimes trainers offered patently bad info
rmation — failing to clarify, for example, that even though teachers were t
o elicit wrong answers from students, they still needed, eventually, to get
to correct ones. Textbooks, too, barely changed, despite publishers’ claims
to the contrary.
With the Common Core, teachers are once more being asked to unlearn an old a
pproach and learn an entirely new one, essentially on their own. Training is
still weak and infrequent, and principals — who are no more skilled at mat
h than their teachers — remain unprepared to offer support. Textbooks, once
again, have received only surface adjustments, despite the shiny Common Cor
e labels that decorate their covers. “To have a vendor say their product is
Common Core is close to meaningless,” says Phil Daro, an author of the mat
h standards.
Continue reading the main story
RECENT COMMENTS
PA 2 hours ago
Amount of resources, Financial or effort that USA can put towards keeping it
s competitive edge, developing nations like India and China...
Nora Madsen 2 hours ago
We needed all this research, international comparisons and Takahashi's life
story to come to the conclusion that American math teachers need...
David Bee 2 hours ago
After reading the article and scores of Comments it was found surprising tha
t no one mentioned (other than Ms. Green, in a way --- see the...
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Left to their own devices, teachers are once again trying to incorporate new
ideas into old scripts, often botching them in the process. One especially
nonsensical result stems from the Common Core’s suggestion that students no
t just find answers but also “illustrate and explain the calculation by usi
ng equations, rectangular arrays, and/or area models.” The idea of utilizin
g arrays of dots makes sense in the hands of a skilled teacher, who can use
them to help a student understand how multiplication actually works. For exa
mple, a teacher trying to explain multiplication might ask a student to firs
t draw three rows of dots with two dots in each row and then imagine what th
e picture would look like with three or four or five dots in each row. Guidi
ng the student through the exercise, the teacher could help her see that eac
h march up the times table (3x2, 3x3, 3x4) just means adding another dot per
row. But if a teacher doesn’t use the dots to illustrate bigger ideas, the
y become just another meaningless exercise. Instead of memorizing familiar s
teps, students now practice even stranger rituals, like drawing dots only to
count them or breaking simple addition problems into complicated forms (62+
26, for example, must become 60+2+20+6) without understanding why. This can
make for even poorer math students. “In the hands of unprepared teachers,”
Lampert says, “alternative algorithms are worse than just teaching them st
andard algorithms.”
No wonder parents and some mathematicians denigrate the reforms as “fuzzy m
ath.” In the warped way untrained teachers interpret them, they are fuzzy.
When Akihiko Takahashi arrived in America, he was surprised to find how rare
ly teachers discussed their teaching methods. A year after he got to Chicago
, he went to a one-day conference of teachers and mathematicians and was per
plexed by the fact that the gathering occurred only twice a year. In Japan,
meetings between math-education professors and teachers happened as a matter
of course, even before the new American ideas arrived. More distressing to
Takahashi was that American teachers had almost no opportunities to watch on
e another teach.
In Japan, teachers had always depended on jugyokenkyu, which translates lite
rally as “lesson study,” a set of practices that Japanese teachers use to
hone their craft. A teacher first plans lessons, then teaches in front of an
audience of students and other teachers along with at least one university
observer. Then the observers talk with the teacher about what has just taken
place. Each public lesson poses a hypothesis, a new idea about how to help
children learn. And each discussion offers a chance to determine whether it
worked. Without jugyokenkyu, it was no wonder the American teachers’ work f
ell short of the model set by their best thinkers. Without jugyokenyku, Taka
hashi never would have learned to teach at all. Neither, certainly, would th
e rest of Japan’s teachers.
The best discussions were the most microscopic, minute-by-minute recollectio
ns of what had occurred, with commentary. If the students were struggling to
represent their subtractions visually, why not help them by, say, arranging
tile blocks in groups of 10, a teacher would suggest. Or after a geometry l
esson, someone might note the inherent challenge for children in seeing angl
es as not just corners of a triangle but as quantities — a more difficult s
tretch than making the same mental step for area. By the end, the teachers h
ad learned not just how to teach the material from that day but also about m
ath and the shape of students’ thoughts and how to mold them.
If teachers weren’t able to observe the methods firsthand, they could find
textbooks, written by the leading instructors and focusing on the idea of al
lowing students to work on a single problem each day. Lesson study helped th
e textbook writers home in on the most productive problems. For example, if
you are trying to decide on the best problem to teach children to subtract a
one-digit number from a two-digit number using borrowing, or regrouping, yo
u have many choices: 11 minus 2, 18 minus 9, etc. Yet from all these options
, five of the six textbook companies in Japan converged on the same exact pr
oblem, Toshiakira Fujii, a professor of math education at Tokyo Gakugei Univ
ersity, told me. They determined that 13 minus 9 was the best. Other problem
s, it turned out, were likely to lead students to discover only one solution
method. With 12 minus 3, for instance, the natural approach for most studen
ts was to take away 2 and then 1 (the subtraction-subtraction method). Very
few would take 3 from 10 and then add back 2 (the subtraction-addition metho
d).
But Japanese teachers knew that students were best served by understanding b
oth methods. They used 13 minus 9 because, faced with that particular proble
m, students were equally likely to employ subtraction-subtraction (take away
3 to get 10, and then subtract the remaining 6 to get 4) as they were to us
e subtraction-addition (break 13 into 10 and 3, and then take 9 from 10 and
add the remaining 1 and 3 to get 4). A teacher leading the “We” part of th
e lesson, when students shared their strategies, could do so with full confi
dence that both methods would emerge.
By 1995, when American researchers videotaped eighth-grade classrooms in the
United States and Japan, Japanese schools had overwhelmingly traded the old
“I, We, You” script for “You, Y’all, We.” (American schools, meanwhile
didn’t look much different than they did before the reforms.) Japanese stu
dents had changed too. Participating in class, they spoke more often than Am
ericans and had more to say. In fact, when Takahashi came to Chicago initial
ly, the first thing he noticed was how uncomfortably silent all the classroo
ms were. One teacher must have said, “Shh!” a hundred times, he said. Late
r, when he took American visitors on tours of Japanese schools, he had to wa
rn them about the noise from children talking, arguing, shrieking about the
best way to solve problems. The research showed that Japanese students initi
ated the method for solving a problem in 40 percent of the lessons; American
s initiated 9 percent of the time. Similarly, 96 percent of American student
s’ work fell into the category of “practice,” while Japanese students spe
nt only 41 percent of their time practicing. Almost half of Japanese student
s’ time was spent doing work that the researchers termed “invent/think.”
(American students spent less than 1 percent of their time on it.) Even the
equipment in classrooms reflected the focus on getting students to think. Wh
ereas American teachers all used overhead projectors, allowing them to focus
students’ attention on the teacher’s rules and equations, rather than the
ir own, in Japan, the preferred device was a blackboard, allowing students t
o track the evolution of everyone’s ideas.
Japanese schools are far from perfect. Though lesson study is pervasive in e
lementary and middle school, it is less so in high school, where the emphasi
s is on cramming for college entrance exams. As is true in the United States
, lower-income students in Japan have recently been falling behind their pee
rs, and people there worry about staying competitive on international tests.
Yet while the United States regularly hovers in the middle of the pack or b
elow on these tests, Japan scores at the top. And other countries now inchin
g ahead of Japan imitate the jugyokenkyu approach. Some, like China, do this
by drawing on their own native jugyokenkyu-style traditions (zuanyan jiaoca
i, or “studying teaching materials intensively,” Chinese teachers call it)
. Others, including Singapore, adopt lesson study as a deliberate matter of
government policy. Finland, meanwhile, made the shift by carving out time fo
r teachers to spend learning. There, as in Japan, teachers teach for 600 or
fewer hours each school year, leaving them ample time to prepare, revise and
learn. By contrast, American teachers spend nearly 1,100 hours with little
feedback.
It could be tempting to dismiss Japan’s success as a cultural novelty, an u
nreproducible result of an affluent, homogeneous, and math-positive society.
Perhaps the Japanese are simply the “math people” Americans aren’t. Yet
when I visited Japan, every teacher I spoke to told me a story that sounded
distinctly American. “I used to hate math,” an elementary-school teacher n
amed Shinichiro Kurita said through a translator. “I couldn’t calculate. I
was slow. I was always at the bottom of the ladder, wondering why I had to
memorize these equations.” Like Takahashi, when he went to college and saw
his instructors teaching differently, “it was an enlightenment.”
Learning to teach the new way himself was not easy. “I had so much trouble,
” Kurita said. “I had absolutely no idea how to do it.” He listened caref
ully for what Japanese teachers call children’s twitters — mumbled nuggets
of inchoate thoughts that teachers can mold into the fully formed concept t
hey are trying to teach. And he worked hard on bansho, the term Japanese tea
chers use to describe the art of blackboard writing that helps students visu
alize the flow of ideas from problem to solution to broader mathematical pri
nciples. But for all his efforts, he said, “the children didn’t twitter, a
nd I couldn’t write on the blackboard.” Yet Kurita didn’t give up — and
he had resources to help him persevere. He went to study sessions with other
teachers, watched as many public lessons as he could and spent time with hi
s old professors. Eventually, as he learned more, his students started to do
the same. Today Kurita is the head of the math department at Setagaya Eleme
ntary School in Tokyo, the position once held by Takahashi’s mentor, Matsuy
ama.
Of all the lessons Japan has to offer the United States, the most important
might be the belief in patience and the possibility of change. Japan, after
all, was able to shift a country full of teachers to a new approach. Telling
me his story, Kurita quoted what he described as an old Japanese saying abo
ut perseverance: “Sit on a stone for three years to accomplish anything.”
Admittedly, a tenacious commitment to improvement seems to be part of the Ja
panese national heritage, showing up among teachers, autoworkers, sushi chef
s and tea-ceremony masters. Yet for his part, Akihiko Takahashi extends his
optimism even to a cause that can sometimes seem hopeless — the United Stat
es. After the great disappointment of moving here in 1991, he made a decisio
n his colleagues back in Japan thought was strange. He decided to stay and t
ry to help American teachers embrace the innovative ideas that reformers lik
e Magdalene Lampert pioneered.
Today Takahashi lives in Chicago and holds a full-time job in the education
department at DePaul University. (He also has a special appointment at his a
lma mater in Japan, where he and his wife frequently visit.) When it comes t
o transforming teaching in America, Takahashi sees promise in individual Ame
rican schools that have decided to embrace lesson study. Some do this delibe
rately, working with Takahashi to transform the way they teach math. Others
have built versions of lesson study without using that name. Sometimes these
efforts turn out to be duds. When carefully implemented, though, they show
promise. In one experiment in which more than 200 American teachers took par
t in lesson study, student achievement rose, as did teachers’ math knowledg
e — two rare accomplishments.
Training teachers in a new way of thinking will take time, and American pare
nts will need to be patient. In Japan, the transition did not happen overnig
ht. When Takahashi began teaching in the new style, parents initially compla
ined about the young instructor experimenting on their children. But his ear
ly explorations were confined to just a few lessons, giving him a chance to
learn what he was doing and to bring the parents along too. He began sending
home a monthly newsletter summarizing what the students had done in class a
nd why. By his third year, he was sending out the newsletter every day. If t
hey were going to support their children, and support Takahashi, the parents
needed to know the new math as well. And over time, they learned.
To cure our innumeracy, we will have to accept that the traditional approach
we take to teaching math — the one that can be mind-numbing, but also comf
ortingly familiar — does not work. We will have to come to see math not as
a list of rules to be memorized but as a way of looking at the world that re
ally makes sense.
CONTINUE READING THE MAIN STORY
598
COMMENTS
The other shift Americans will have to make extends beyond just math. Across
all school subjects, teachers receive a pale imitation of the preparation,
support and tools they need. And across all subjects, the neglect shows in s
tudents’ work. In addition to misunderstanding math, American students also
, on average, write weakly, read poorly, think unscientifically and grasp hi
story only superficially. Examining nearly 3,000 teachers in six school dist
ricts, the Bill & Melinda Gates Foundation recently found that nearly two-th
irds scored less than “proficient” in the areas of “intellectual challeng
e” and “classroom discourse.” Odds-defying individual teachers can be fou
nd in every state, but the overall picture is of a profession struggling to
make the best of an impossible hand.
Most policies aimed at improving teaching conceive of the job not as a craft
that needs to be taught but as a natural-born talent that teachers either d
ecide to muster or don’t possess. Instead of acknowledging that changes lik
e the new math are something teachers must learn over time, we mandate them
as “standards” that teachers are expected to simply “adopt.” We shouldn’
t be surprised, then, that their students don’t improve.
Here, too, the Japanese experience is telling. The teachers I met in Tokyo h
ad changed not just their ideas about math; they also changed their whole co
nception of what it means to be a teacher. “The term ‘teaching’ came to m
ean something totally different to me,” a teacher named Hideto Hirayama tol
d me through a translator. It was more sophisticated, more challenging — an
d more rewarding. “The moment that a child changes, the moment that he unde
rstands something, is amazing, and this transition happens right before your
eyes,” he said. “It seems like my heart stops every day.”